PHYSICAL REVIEW E 87, 042134 (2013)

Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with threesimulation methods

Michael Engel,1 Joshua A. Anderson,1 Sharon C. Glotzer,1,* Masaharu Isobe,2,3 Etienne P. Bernard,4 and Werner Krauth5,†1Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

2Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, 466-8555, Japan3Department of Chemistry, University of California, Berkeley, California 94720, USA

4Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA5Laboratoire de Physique Statistique, Ecole Normale Superieure, UPMC, CNRS, 24 Rue Lhomond, 75231 Paris Cedex 05, France

(Received 9 November 2012; published 30 April 2013)

We report large-scale computer simulations of the hard-disk system at high densities in the region of themelting transition. Our simulations reproduce the equation of state, previously obtained using the event-chainMonte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and withevent-driven molecular dynamics. We analyze the relative performance of these simulation methods to sampleconfiguration space and approach equilibrium. Our results confirm the first-order nature of the melting phasetransition in hard disks. Phase coexistence is visualized for individual configurations via the orientational orderparameter field. The analysis of positional order confirms the existence of the hexatic phase.

DOI: 10.1103/PhysRevE.87.042134 PACS number(s): 05.70.Fh, 64.70.dj, 61.20.Ja, 33.15.Vb

I. INTRODUCTION

The phase behavior of hard disks is one of the oldest andmost studied problems in computational statistical mechanics.It inspired the use of Markov-chain Monte Carlo [1] as wellas molecular dynamics [2]. Important progress in understand-ing hard-disk melting [3–5] was made recently. Using theevent-chain Monte Carlo algorithm (ECMC) [6], a first-orderliquid-hexatic transition was identified [7]. This transitionfrom the low-density phase to an intermediate phase precedesa continuous hexatic-solid transition, and thus the liquidtransforms to a solid through an intermediate hexatic phase.

Controversy concerning the nature of hard-disk melting haspersisted for decades. Indeed, the recently discovered first-order liquid-hexatic melting transition differs from the stan-dard Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY)scenario [8–13], which predicts continuous transitions bothfrom the liquid to the hexatic and from the hexatic to the solid.It is also at variance with the first-order liquid-solid transitionscenario, which exhibits no intermediate hexatic phase andhas been much discussed [3,13–18]. Near the critical density,the system is correlated across roughly a hundred disk radii,and the hard-disk liquid-hexatic transition is, thus, weaklyfirst-order [7], with only a small discontinuity in density at thetransition.

For several decades, the algorithms used for this problem[12,13,18–20] were unable to equilibrate systems sufficientlylarger than the spatial correlation length to reliably investigatethe existence and the nature of the hexatic phase. This was oneorigin for the controversy surrounding this problem. Anotherreason was that the manifestation of a first-order transitionin the NV T ensemble, and in particular the fundamentaldifference between a van der Waals loop and a Mayer-Wood loop indicating equilibrium phase coexistence, was not

*sglotzer@umich.edu†werner.krauth@ens.fr

universally accepted in the hard-disk community, although ithad been clearly discussed in the literature [21–23].

Here, we complement and compare the recent event-chainresults with a massively parallel implementation of the localMonte Carlo algorithm (MPMC) [24] and with event-drivenmolecular dynamics (EDMD) [25]. These methods provide uswith by far the largest independent data sets ever acquired forthe hard-disk melting transition. Our simulations reproduceto very high precision the equation of state of Ref. [7],illustrating phase separation. To characterize the nature of thetwo hard-disk phase transitions, we graphically represent theorientational and positional order parameter fields and analyzepositional correlation functions.

II. SIMULATION METHODS

A. System definition

We consider a system of N hard disks of radius σ in asquare box of size L × L. The phase diagram of the systemdepends only on the density (packing fraction) η = Nπσ 2/L2,as the pressure is proportional to the temperature T . Thedimensionless pressure is given by

P ∗ = (2σ )2

m⟨v2

x

⟩P = βP (2σ )2, (1)

with the inverse temperature β, mass m, and the velocityalong one axis vx . All simulations are conducted in the NV T

ensemble and, although our algorithms differ in the waythey evolve the system, they all sample the same equilibriumprobability distribution in configuration space. At finite N ,the equilibrium phase coexistence that we will observe isspecific to this ensemble and absent, for example, in the NPT

ensemble. As for any model with short-range interactions, thethermodynamic limit is independent of the ensemble.

B. Algorithms and implementations

Local Monte Carlo (LMC) has been a popular simulationmethod for hard disks [1,13,18,20]. At each time step, one

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MICHAEL ENGEL et al. PHYSICAL REVIEW E 87, 042134 (2013)

random disk is selected and a trial move is applied to it.The move is accepted unless it results in an overlap withanother disk. LMC is both relatively inefficient in samplingconfiguration space and inherently serial, limiting the sizefor which the system can be brought to equilibrium at highdensities η ∼ 0.7 to about N ∼ 105 particles (see Ref. [26]for a basic discussion). Our alternative approaches utilizemodern computer resources more efficiently and equilibratethe system faster. They also provide independent checks of theequilibrium phase behavior. LMC is used for comparison andas a reference to previous work.

Massively parallel Monte Carlo (MPMC) [24] is aparallel extension of LMC. It again applies a local trialmove but maximizes the number of simultaneous updates.MPMC extends the stripe decomposition method [27] to amassive number of threads using a four-color checkerboardscheme [28]. By placing disks into cells of width w � 2σ ,concurrent threads execute over one out of four subsets ofcells in parallel. Within each cell, particles are chosen fortrial moves in a shuffled order. The number of trial moves isfixed independent of cell occupancy. Trial moves that woulddisplace disks across cell boundaries are rejected. The order ofthe four checkerboard subsweeps is also sampled as a randompermutation. In this manner, an entire sweep over N particlessatisfies detailed balance. The reverse sweep corresponds to aninverse shuffling and cell sequences with opposite trial movesand occurs with equal probability. To ensure ergodicity, the cellsystem is randomly shifted after each sweep. We implementMPMC on a graphics processing unit (GPU) using CUDA.Details are found in Ref. [24]. The MPMC simulations executesimultaneously on all 1536 cores of a NVIDIA GeForceGTX 680.

In event-driven molecular dynamics (EDMD), individualsimulation events correspond to collisions between pairs ofdisks [2]. The simulation is advanced sequentially from onecollision event to the next. Between collisions, disks move atconstant velocity (see Ref. [26] for a basic discussion). Thecomputation of future collisions and the update of the eventschedule are performed efficiently using a binary tree andrelating searching schemes [29–32]. As a result, one collisionevent in EDMD costs only about 10–20 times more CPU timethan a LMC trial move, even for large system sizes. EDMDdrives the system quite efficiently through configuration spaceand clearly outperforms LMC. The simulations with thisalgorithm use an Intel Xeon E5-1660 CPU with a clock speedof 3.30 GHz.

Event-chain Monte Carlo (ECMC) [6] replaces individualtrial moves by a chain of collective moves that all translateparticles in the same direction. At the beginning of each MonteCarlo move, a random starting disk and a move directionare selected. The starting disk is displaced in the chosendirection until it collides with another disk. This new diskis then displaced in the same direction until another collisionoccurs or until the lengths of all displacements add up to atotal distance, an internal parameter of the algorithm whichis typically chosen such that the chain consists of ∼√

N

disks. With periodic boundary conditions, ECMC is free ofrejections. Global balance and ergodicity are preserved bymoving in two directions only, for example, to the right andup. ECMC is faster than LMC and EDMD [6]. The simulations

with this algorithm were performed on an Intel Xeon E5620CPU with a clock speed of 2.40 GHz.

For the large systems considered in this study, the ratiobetween the large absolute particle coordinates and the poten-tially small interparticle distances becomes comparable to theaccuracy of single floating point precision, so that cancellationerrors become critical. Different strategies allow us to copewith this problem. MPMC performs all computations in single-precision because today’s GPUs run significantly slowerin double-precision. We mitigate floating-point cancellationerrors by placing each particle in a coordinate system local toits cell. In this way, differences of relative positions are lessaffected by floating point precision than absolute positions.This strategy was also applied in Ref. [7]. EDMD calculationsare performed in double-precision to fully resolve multiplecoincident collisions and to span the entire time domain fromindividual collision times to total simulation time. The ECMCalgorithm is implemented for this work in double-precision.We use this implementation to derive high-precision numbersat the density η = 0.698. Data points for ECMC at otherdensities are taken from Ref. [7], which employs single-precision. The comparison of single-precision and double-precision calculations at η = 0.698 indicates that the twoversions of ECMC yield the same result for the pressure.

C. Pressure computation

The complete statistical behavior of hard disks is containedin the equation of state (pressure versus volume or density),which requires the precise evaluation of the internal pressure ofthe system. The equation of state allows computing the interfa-cial free energy and tracking the changes in the geometry of co-existing phase regions in a finite system (see Refs. [7,21–23]).

In the NV T ensemble, the pressure is a dependentobservable. In Monte Carlo, it has to be computed fromstatic configurations, while in EDMD, it may in addition bederived from the collision rate. The disparity of our approachesto calculate pressure constitutes one more check for theimplementations of our algorithms.

1. Pressure from static configurations

In systems of isotropic particles with pairwise interactions,the pressure can be computed from static configurationsthrough the pair-correlation function g(r) [1]. The functiong(r) is defined as the distribution of particle pairs at distancer = |xi − xj |, normalized such that g(r → ∞) = 1. In prac-tice, particle distances are binned into a histogram with binsize δr . If n out of p pair distances are found to lie in theinterval [r − δr/2,r + δr/2], then we have

g(r) = n/p

2πrδr/V. (2)

The pressure P = −(∂F/∂V )T ,N is calculated from thefree energy F = −β−1 ln Z and the partition function

Z = 1

N !

∫ L

0· · ·

∫ L

0dx1 · · · dx2Nθ (x1 · · · x2N )

= V N

N !

∫ 1

0· · ·

∫ 1

0dα1 · · · dα2Nθ (α1 · · ·α2N ), (3)

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4.5

5

5.5

6

6.5

7

0 0.02 0.04 0.06 0.08 0.1

g(r)

(r−2σ)/σ

g(r)fit

(a)

−2.10−3

0

2.10−3

0 0.1

g(r)

-fit

(r−2σ)/σ

single precision

(b)

−2.10−4

0

2.10−4

0 0.1

g(r)

-fit

(r−2σ)/σ

double precision

(c)

FIG. 1. (Color online) (a) Pair-correlation function g(r) close tocontact for N = 5122 at density η = 0.698 using LMC. Error bars arecomputed through 64 independent simulations. g(r) is fitted with afourth-order polynomial. Difference between g(r) and the polynomialfit with the single-precision data (b) and double-precision data (c) forthe histogram.

where αj = xj/L are the particle coordinates relative to thesimulation box. The Boltzmann weight θ is the characteristicfunction for overlap, i.e., zero if the configuration containsoverlaps and one otherwise. A change of volume leaves theα unchanged but rescales the positions and the pair distances.θ (α1 · · · α2N ) is only affected if one of the pair distances is atcontact, hence

βP = N

V+ σ

V

⟨∂θ

∂r

⟩∣∣∣∣r=2σ+

= N

V[1 + 2η g(2σ+)]. (4)

To access the contact value of g(r), we fit the histogram ofpair distances obtained from LMC by a polynomial as shown inFig. 1 and then extrapolate the fit to r = 2σ from the right. Wechoose the bin size as δr = 10−3σ . The histogram is limitedto r ∈ [2σ,2.1σ ], and the fit is performed with a fourth-orderpolynomial. These parameters are sufficient to obtain a relativesystematic error of less than 10−5. In the single-precisionversion of the algorithm, g(r) shows correlated fluctuationswhich lead to systematic errors in the value of g(r) forany single bin [see Figs. 1(b) and 1(c)]. These errors areonly due to floating point round-off of the pair-correlationfunction (the sampled configurations are essentially the same).However, they are periodic with zero mean and do not affectthe fit significantly. We verified that single-precision roundinginduces a relative systematic error smaller than 10−5 on theestimated value of g(2σ+).

2. Dynamic pressure computation

In molecular dynamics, static configurations can be an-alyzed as before, but the pressure is computed directlyand more efficiently from the collision rate via the virialtheorem [33–35]. This avoids binning and extrapolations. Thenondimensional virial pressure in two dimensions is given by

βP = N

V

[1 − βm

2ttot

1

N

∑collisions

bij

], (5)

TABLE I. Test of the pressure computations for N = 2562 atη = 0.698. The table lists for each algorithm the number of runs,number of displacements (Disp.) per run, pressure, and standard error.Results of all four algorithms agree within their numerical accuracies.

Algorithm Runs Disp./run βP (2σ )2 Std. error

LMC 64 6 × 1011 9.170 46 1.5 × 10−4

EDMD 100 1010 9.170 76 1.8 × 10−4

ECMC 32 5 × 1011 9.170 62 8.7 × 10−5

MPMC 8 6 × 1013 9.170 78 4.5 × 10−5

where ttot is the total simulation time. The collision force bij =rij · vij is defined between the relative positions and relativevelocities of the collision partners. In equilibrium, the averagevirial for hard disks equals [36]

〈bij 〉 = −2σ

√π

βm. (6)

Therefore, the pressure is simply given by the collision rate = 1/t0, the reciprocal of the mean free time t0, as

βP = N

V

[1 + σ

√πβm

2

]. (7)

To test the pressure computations, we compute the pressurewith all algorithms at one representative state point. With eachalgorithm, we perform between 8 and 100 independent runsto compute the statistical standard error. Results obtained withthe four algorithms agree within numerical accuracy to �10−4,which is sufficient for our purposes (see Table I).

III. PERFORMANCE COMPARISON

One of the slowest processes during the time evolution ofthe hard-disk system at high density is the fluctuation of theglobal orientation order parameter

�6 = 1

N

∑j

ψj ,

which is the spatial average of the local orientational orderparameter

ψj = 1

6

6∑k=1

exp(i 6φj,k). (8)

The sum is over the six closest neighbors k of disk j ,and φj,k is the angle between the shortest periodic vectorequivalent to xk − xj and a chosen fixed reference vector. Thisapproach is simpler than using the Voronoi construction [7],without affecting the autocorrelation functions. To determinethe efficiency of our algorithms, we track the autocorrelationfunction of �6 [6],

C(�t) = 〈�6(t)�∗6 (t + �t)〉t

〈|�6|2〉 . (9)

The �6 → �6 + π symmetry in the square box imposes thatC(�t) decays to zero for infinite times. In the asymptotic limit,the decay is exponential, C(�t) ∝ exp(−�t/τ ), and we obtainthe correlation time τ from a fit of the pure exponential part(see Fig. 2).

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0.01

0.1

1

0 200000 400000

C(Δ

t)

Δt (disp.)

MPMCLMC

ECMCEDMD

(a)

0.01

0.1

1

0 2 4 6C

(Δt)

Δt (hours)

LMCEDMDECMCMPMC

(b)

FIG. 2. (Color online) Autocorrelation function of the globalorientation order parameter �6(t) for N = 5122, η = 0.698 obtainedwith LMC, EDMD, ECMC, and MPMC. (a) Time is measured innumber of attempted displacements (or collisions) per disk. (b) Timeis measured in CPU or GPU hours.

We compare speeds for N = 5122 at η = 0.698, that is,in the dense liquid close to the liquid-hexatic coexistence.Although still a liquid, the correlation length of the localorientational order parameter ψj at this density is ∼50σ . Sucha large correlation length induces a long correlation time, andequilibration requires >106 trial moves per disk for LMC. Forthe test, each algorithm is set to its optimal internal parameters.Total simulation run times are on the order of 103τ to 104τ .

Figure 2 illustrates that the autocorrelation functions decayroughly as pure exponentials. The time unit in Fig. 2(a)corresponds to the number of attempted displacements perdisk (number of collisions in the case of EDMD and ECMC).As expected, MPMC decays slightly more slowly than LMCbecause trial moves across cell boundaries are rejected. ECMCand EDMD are significantly faster than LMC, confirming thatthese two methods sample configuration space more efficientlybut slower than MPMC. The time unit in Fig. 2(b) correspondsto the real simulation time (CPU or GPU time).

Correlation times, number of attempted displacements perhour, and accelerations with respect to LMC are summarized inTable II. EDMD and ECMC sample configuration space moreefficiently by a factor of 39 and 28 times, respectively, whileMPMC samples configuration space slightly less efficientlyby a factor of 0.9. Evidently, the efficiency of the simpleLMC algorithm is improved significantly with speed-ups of10, 70, and 320 for EDMD, ECMC, and MPMC, respectively.

TABLE II. Speed comparison of the four hard-disk algorithms forN = 5122, η = 0.698. The correlation time τ is measured in numberof displacements (or collisions) per disk. Disp./hr represents thenumber of displacements per hour achieved in our implementations.The two rightmost columns show the speed-up of the algorithms innumber of displaced disks and in terms of CPU or GPU time incomparison to LMC.

Algorithm τ/disp. Disp./hr τLMC/τ Speed-up

LMC 7 × 105 6.5 × 109 1 1EDMD 1.8 × 104 1.7 × 109 39 10ECMC 2.5 × 104 1.6 × 1010 28 70MPMC 8 × 105 2.3 × 1012 0.9 320

Although speeds in Table II correspond to somewhat differenthardware, as indicated under Methods, the numbers give aclear idea of practical improvements that can be obtained withrespect to LMC.

IV. RESULTS AND DISCUSSION

A. Equation of state at high density

In their seminal work, Alder and Wainwright observed aloop in the equation of state of hard disks [3]. As explainedby Mayer and Wood [21] (see also Refs. [22,23]), this loopis a result of finite simulation sizes and, therefore, differsconceptually from a classic van der Waals loop, which isderived in the thermodynamic limit. The branches of theMayer-Wood loop are thermodynamically stable but vanishin the limit of infinite size.

It is known that the presence of a Mayer-Wood loop in theequation of state is observed in systems showing a first-ordertransition as well as systems showing a continuous transition[37]. However, the behavior of these loops with increasingsystem size is different. For a first-order transition, the loop ispresent in the coexistence region and is caused by the interfacefree energy �F . At a given density, the interface free energyper disk, �f = �F/N , can be computed by integrating theequation of state [7]. In two dimensions, it scales as �f ∝N−1/2. In contrast, for a continuous transition, �f decaysfaster, normally such that �F is constant, that is �f ∝ N−1,and the equation of state becomes monotonic for large enoughsystems. The scaling of �f with system size, together witha fixed finite separation of the peaks for large system sizes,is a reliable indicator of the first-order character of a phasetransition [38].

Figure 3 shows the equation of state for N = 2562 andN = 5122, obtained with ECMC, EDMD, and MDMC. Errorbars (standard errors) are computed through independent simu-lations. The inset shows the relative pressure difference �P/P

of EDMD and MDMC with ECMC. Error bars correspondagain to the standard error on �P/P . We observe that the

9.17

9.175

9.18

9.185

9.19

9.195

9.2

0.7 0.705 0.71 0.715 0.72

Pres

sure

βP(2

σ)2

Density η

N = 2562, ECMCEDMDMPMC

N = 5122, ECMCEDMDMPMC

−10−40

10−4

0.7 0.71 0.72

ΔP/P

η

FIG. 3. (Color online) Equation of state from ECMC, EDMD, andMPMC for N = 2562 and N = 5122. Error bars are mostly smallerthan the symbols. Results agree within one standard deviation. Theinset shows the relative pressure difference �P/P of EDMD andMPMC with respect to ECMC for N = 2562.

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9.17

9.175

9.18

9.185

9.19

9.195

9.2

0.7 0.705 0.71 0.715 0.72

Pres

sure

βP(2

σ)2

Density η

N = 10242, ECMCMPMC

−10−40

10−4

0.7 0.71 0.72ΔP

/Pη

FIG. 4. (Color online) Equation of state from ECMC and MPMCfor N = 10242. Error bars are mostly smaller than the symbols.Results agree within one standard deviation. The inset shows therelative pressure difference �P/P of MPMC with respect to ECMC.

three independent simulations agree. In a similar way, Fig. 4compares results from ECMC and MPMC for N = 10242. Forthis system size and the currently available computer hardware,equilibration with EDMD takes too long to be practical. Again,all results agree within standard deviations. Note that while theMayer-Wood loop shrinks with system size, the position of thelocal extrema stay fixed close to η = 0.702 and η = 0.714.

B. Orientational order parameter field

The degree and distribution of local order in a system canbe analyzed with the help of order parameters. By averaginga given order parameter attached to each particle over a smallsampling area surrounding the particle, we obtain a continuousfunction, the corresponding order parameter field. Typicalsampling areas used in this work contain between 5 and 20disks.

To show the separation of the liquid and the hexatic phase,we graphically represent the orientational order parameter fieldψ(x) for configurations at densities where coexistence occurs.As for any first-order phase transition in two dimensions,the characteristic geometry of the region of the minorityphase with increasing density is expected to change from anapproximately circular bubble into a parallel stripe and againinto a circular bubble. Reference [7] used the projection of thelocal orientational order ψk on the global orientational order�6 and a linear color code. This projection is not a uniquemeasure for the orientational order parameter field. Instead, inFig. 5, we use a circular color code on data obtained by longMPMC simulations with N = 10242.

We observe that the system is uniformly liquid at η = 0.700and the color fluctuates on the scale of the correlation length.At η = 0.704, the representations indicate the presence of acircular bubble of hexatic phase, visible as a large region ofconstant purple color, whereas at η = 0.708 a stripe minimizesthe interfacial free energy. The hexatic phase is now visible inblue, while the liquid is characterized by fluctuations of thedirection of �6. The constant color in the region of the hexaticphase confirms the presence of the same orientational order

(a) (b)

(c)

0

500

1000

1500

2000

r/σ

0◦

45◦90◦

135◦

180◦

225◦

270◦315◦

ψ-phase

(d)

FIG. 5. (Color online) Orientational order parameter field ψ(x)of configurations obtained with the MPMC algorithm for system sizeN = 10242. With increasing density, (a) pure liquid (η = 0.700), (b) abubble of hexatic phase (η = 0.704), and (c) a stripe regime of hexaticphase (η = 0.708) are visible. The interface between the liquid andthe hexatic phase is extremely rough. (d) A scale bar illustrates thesize of the fluctuations. The phase of ψ is represented via the colorwheel.

across the system. Additional important evidence for the natureof the transition is provided by the spatial correspondencebetween variations in local density and orientational order,which are included as a movie in the supplemental material ofthis work [39]. The movie illustrates for the density η = 0.71that the system is ergodic by seeing the patches of the twophases appear and disappear at different locations but withroughly fixed ratio of areas.

C. Positional order parameter field

To identify the structure of the ordered phase in coexistencewith the liquid, we analyze the positional order in the system.The goal is to distinguish the hexatic phase, which has short-range positional order (characterized by exponential decayof the correlation function) and quasi-long-range orientationalorder (algebraic decay), from the two-dimensional solid, whichhas quasi-long range positional order [40] and long-rangeorientational order (no complete decay).

In Ref. [7], the positional order was analyzed using thetwo-dimensional pair-correlation function in direct space andthe decay of the positional correlation function at the wavevector q0 corresponding to the maximum value of the firstdiffraction peak of the structure factor

S(q) = 1

N

∑n,m

exp[iq · (xn − xm)]. (10)

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MICHAEL ENGEL et al. PHYSICAL REVIEW E 87, 042134 (2013)

(a) (b)

(c) (d)

FIG. 6. (Color online) Positional order parameter field χ (x) ofconfigurations obtained with (a), (b) the MPMC and (c), (d) ECMCalgorithms for system size N = 10242. (a), (c) In the hexatic phase(η = 0.718), positional order is short-range. (b), (d) Toward higherdensity (η = 0.720), fluctuations are much weaker and bounded, asexpected for a continuous transition to a solid phase. The scale barand the color code for the phase of χ are identical to the scale bar andthe color code in Fig. 5.

With this classic method, the wave vector q0 must be chosencarefully to correspond to a diffraction peak. In some previousworks [13,41], it was assumed that q0 would correspond tothe reciprocal vector of a perfect triangular lattice of edgelength a0, namely |q0| = 2π/(a0

√3/2). This assumption is

not correct because the solid phase has a finite density ofvacancies and other defects in equilibrium, which increasesthe effective lattice constant [7].

We visualize the positional order parameter field χ (x)calculated from the positional order parameter

χj = exp(iq0 · xj ), (11)

slightly above the upper critical density of coexistence withthe liquid (Fig. 6). At η = 0.718, the system resembles apatchwork of independent, solid-like regions of size in theorder of a few hundred σ . Some regions show almost constantχ , while others are characterized by regular interferencefringes (oscillatory waves) with a fixed wave vector. It can beshown that each end of a fringe corresponds to one unpaireddislocation [44]. We find that fringes can be made to disappearseparately in each region by small rotations of q0 aroundthe origin. This behavior is consistent with the existence ofsmall-angle grain boundaries separating neighboring solid-likeregions, exactly as predicted by the KTHNY scenario [14].Note that if fringes disappear on one side of a boundary

separating two regions, they necessarily have to reappear onthe other side with the sequence of the colors reversed.

On physical grounds, the hexatic phase is not expected tobe stable up to close packing. Indeed, already at η = 0.720,the positional order field shown in Fig. 6 fluctuates muchmore slowly and is highly correlated throughout the system asexpected for a solid phase.

D. Positional correlation function

The decay of positional order can be analyzed using thepositional correlation function in reciprocal space,

Cq0 (r) = 〈exp[iq0 · (xn − xm)]〉 . (12)

The averaging for Cq0 (r) is done on two levels. First, weaverage over neighboring pairs that satisfy |xn − xm| ∈ [r −σ,r + σ ]. In addition, we conduct an average over independentconfigurations, which can be a time average or an ensembleaverage. For details on the configuration averaging see theAppendix. As the system can perform global rotations duringthe simulations, q0 rotates from one configuration to the other.Each configuration is individually rotated so that �6 is alignedin the same direction for all of them. We verified that alignmenterrors are sufficiently small and can be neglected. As a resultof the configuration average, finite-size effects present at largedistances r in the form of interferences are suppressed.

Figure 7 shows Cq0 (r) at η = 0.718 and η = 0.720. Theresults of ECMC and MPMC are again in good agreement.We observe that Cq0 (r) decays exponentially at η = 0.718.Therefore, η = 0.718 cannot be in the solid phase. The lengthscale of the exponential decay is in the order of 100σ , whichcorresponds approximately to the width of the interferencefringes in Fig. 6. Since the coexistence phase ends at η �0.716, the region η � 0.716 is thus hexatic. We have shownonce more that the first-order transition observed in Fig. 3connects a liquid and a hexatic phase.

The positional order increases drastically at η = 0.720.Cq0 (r) decays almost as a power law, r−1/3, which is thestability limit for the solid phase in the KTHNY theory. Thus,the stability regime of the hexatic phase comprises a narrow

0.001

0.01

0.1

1

10 100 1000

Cq 0

(r)

r/σ

∝r−1/3

η = 0.720, ECMCMPMC

η = 0.718, ECMCMPMC

∝exp(−r/120σ)

(a)

0.001

0.01

0.1

1

10 100 1000

Cq 0

(r)

r/σ

∝r−1/3

η = 0.720, ECMCMPMC

η = 0.718, ECMCMPMC

∝exp(−r/100σ)

(b)

FIG. 7. (Color online) The positional correlation function Cq0 (r)shows exponential decay at density η = 0.718 and approaches apower law ∝r−1/3 at η = 0.720. (a) System size N = 5122: Excellentagreement between ECMC and MPMC. (b) System size N = 10242:Excellent agreement in the hexatic phase (η = 0.718) and fairagreement at the approach of the solid phase (η = 0.720). Ouralgorithms fall out of strict equilibrium in the solid and long-scalecorrelations become sensitive to the boundary conditions.

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0.001

0.01

0.1

1

10 100 1000

Cq 0

(r)

r/σ

n = 1248

1632

FIG. 8. (Color online) Influence of the configuration averagingon the positional configuration functions in the hexatic phase. Forsingle configurations (n = 1, data incoherently averaged), positionalcorrelations cannot decay below a level given by the square root ofthe ratio of the correlated domain size to the system size. Coherentaveraging over n = 2, 4, 8, 16, and 32 configurations reduces thenoise level and the correlations. The data corresponds to a single longMPMC run with N = 10242 particles at density η = 0.718.

range of density. An additional characterization of the hexaticphase with ECMC including a study of the diffraction peakshape can be found in Ref. [43] and in the supplementalmaterial of Ref. [7]. The continuous transformation to thesolid phase and the nature of the hexatic phase agree with theKTHNY scenario.

Slight variations in the positional correlations can beobserved in Fig. 7 at density η = 0.720 for distances com-parable to the system size. Two factors play a role. First, therelaxation becomes very slow at the onset of the solid phase.Our largest system no longer achieves full global rotationswith respect to �6. Second, the positional correlations spanthe whole simulation box and, therefore, depend slightly onthe orientation of the crystal. Longer and larger simulationsare necessary to determine the location of the hexatic-solidtransition with good precision. However, the general absenceof a loop in pressure is sufficient to rule out a first-orderhexatic-solid transition. We note that while our simulationsfall out of strict equilibrium with respect to global rotations inthe solid phase, they remain fully ergodic within our simulationtimes on both sides of the liquid-hexatic transition.

V. CONCLUSION

We analyzed the thermodynamic behavior of the hard-disksystem close to the melting transition using independentimplementations of three different simulation algorithms tosample configuration space and two distinct approaches for thepressure computation. The equation of state data of Ref. [7]are confirmed within numerical accuracy both qualitatively

and quantitatively. Typical relative errors are �10−4, morethan one order of magnitude smaller than finite-size effectsfor systems with up to N = 10242 particles. Such finite-sizeeffects are manifested in the form of a Mayer-Wood loop in theequation of state. Our analysis of orientational and positionalorder parameters confirms the presence of a first-order phasetransition from liquid order to hexatic order and a continuousphase transition from hexatic order to a solid phase.

ACKNOWLEDGMENTS

M.E., J.A.A., and S.C.G. acknowledge support by theAssistant Secretary of Defense for Research and Engineering,US Department of Defense No. N00244-09-1-0062. M.I. isgrateful for financial support from the CNRS-JSPS ResearcherExchange Program for staying at ENS-Paris and Grant-in-Aid for Scientific Research from the Ministry of Education,Culture, Sports, Science and Technology No. 23740293. W.K.acknowledges the hospitality of the Aspen Center for Physics,which is supported by the National Science Foundation GrantNo. PHY-1066293. MPMC simulations were performed on aGPU cluster hosted by the University of Michigan’s Centerfor Advanced Computing. EDMD simulations were partiallyperformed using the facilities of the Supercomputer Center,ISSP, University of Tokyo, and RCCS, Okazaki, Japan. Weacknowledge helpful correspondence and discussions with B.J.Alder, D. Fiocco, S.C. Kapfer, and S. Rice.

APPENDIX: ROLE OF CONFIGURATION AVERAGING

It is instructive to analyze the influence of configurationaveraging that we employ to calculate the correlation func-tion Cq0 (r). We define a configuration average as the timeaverage or the ensemble average of Cq0 (r) over individualconfigurations. This coherent averaging procedure stands incontrast to the incoherent averaging procedure, where we binthe function Cq0 (r), determine the maximum within each ofthe bins, and average the maxima. As shown in Fig. 8, forincoherently averaged single configurations the long-distancecorrelations do not decay below a sampling threshold that isset by the inverse square root of the number of independentdomains in the sample. For our large system of N = 10242

particles, given that the sample size is L/2 ∼ 1000σ and thepositional correlation length is in the order of 100σ , there areabout 100 independent domains. Residual correlations visiblein the figure as a plateau at large values of r/σ correspond towhat would be expected from about 100 independent randomlypositioned (yet equally oriented) lattices. As illustrated inthe figure, coherent averaging over oriented configurationsincreases the effective number of independent domains, thus,to reduce the noise level. The same effect would be obtainedif we could equilibrate yet larger systems.

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